## Relativistic two spinning body problem solved exactly for the first time

Using a special technique recently developed for celestial mechanics and practically unknown in Relativity, ACT researchers have analyzed and solved exactly, for the first time, the problem of motion of two orbiting spinning bodies considering all the most relevant relativistic corrections.

One of the most important applications of General Relativity is the effect that curved spacetime has on the bodies of our closest celestial neighborhood: the Solar System and, more in general on a system of gravitating objects. The first researchers working on relativity, Einstein included, were very much aware of this fact and dedicated much time to the understanding of this issue, exploring the so-called Post-Newtonian limit of General Relativity.

Nowadays the importance of the Post-Newtonian (PN) approximation has grown even beyond its original purpose. Space technologies enable us to make very precise measurements of the dynamics of many of the planets and we can observe directly the deviations form Newtonian celestial mechanics. In addition, the PN approximation is a key to the understanding of the mechanisms behind the generation of gravitational waves by astrophysical object and constitutes, as consequence, an important tool to probe gravitation in the strong filed regime. On a more technological point perspective the PN approximation is also a keystone in the construction of any realistic relativistic positioning system (i.e. a positioning system based completely on Einstein Gravity, rather than Newtonian mechanics).

In this perspective it is important to develop new techniques that allow an easier treatment of the problem of Post-Newtonian gravitation. The ACT researchers have worked exactly in this direction. More specifically, they used a relatively new technique in Hamiltonian perturbation theory, that is based on the so-called Lie series. Such technique makes it possible to simplify the calculations necessary for the resolution of a problem of motion for a physical which can be expressed as the sum of an exactly solvable system and some perturbative terms.

The ACT researchers have therefore applied Lie perturbation theory to solve the so called "1PN restricted two spinning body problem" i.e. the problem of motion of two gravitating point masses that: (i) posses an intrinsic rotation (spin), (ii) are such that one mass in much bigger than the other (reduced problem), (iii) take in account the most important (1PN) relativistic corrections.

Lie perturbation theories not only allows a thorough investigation of the details of the motion of the two objects, but also to find a exact solution for the system. The derivation of this solution opens the way to a important series of applications that range from the measurement of relativistic effects in the Solar System via the data that will come form Bepi-Colombo probe to the derivation of new test for relativity based for example on the orbital behavior of exoplanets.

The idea of using Lies series to treat the Post-Newtonian limit of General Relativity is the result of the multidisciplinary approach to research which is typical of ACT projects: not many people working in the field of celestial mechanics would consider using Lie perturbation theory on the problem of PN gravitation, and the majority of the researchers in relativity are unaware of the possibility of solving perturbative Hamiltonian problems via Lie series.

Link to the article **
"A first-order secular theory for the post-Newtonian two-body problem with spin-I. The restricted case" ** can be found here.

Researchers:

Sante Carloni

Francesco Biscani